Stanisław Leśniewski

Stanisław Leśniewski (1886–1939) was one of the
principal founders and movers of the school of logic that flourished in
Warsaw between the two world wars. He was the originator of an
unorthodox system of the foundations of mathematics, based on three
formal systems: Protothetic, a logic of propositions and their
functions; Ontology: a logic of names, and functors of arbitrary order;
and Mereology, a general theory of part and whole. His concern for
utmost rigor in the formalization and execution of logic, coupled with
a nominalistic rejection of abstract entities, led to a precise but
highly unusual metalogic. His strictures on correctly
distinguishing use from mention of expressions, his canons of correct
definition, and his mereology, have all informed the logical
mainstream, but the majority of his logical views and innovations have
not been widely adopted. Despite this, his influence as a teacher and
as a motor for logical innovation are widely acknowledged. He remains
one of logic's most original figures.

Stanisław Leśniewski was born on 28 March 1886 at
Serphukhov, near Moscow, to Izydor, an engineer working on the
construction of the Trans-Siberian Railway, and Helena, née
Palczewska. He was baptised at St. Stanislav's church in
St. Petersburg. His mother died when he was young and his father
remarried. He attended classical Gimnazjum (grammar school
studying classical languages) in Irkutsk in Siberia. Between 1904 and
1910 Leśniewski studied Philosophy and Mathematics in Germany,
Switzerland and Russia, at Leipzig, Zurich, Heidelberg,
St. Petersburg, and at Munich, where he heard lectures by Hans
Cornelius, Moritz Geiger and Alexander Pfänder. In 1910 he went
as a doctoral student to Lwów University, then in
Austria-Hungary, where the foremost Polish philosopher of the day,
Kazimierz Twardowski, a student of Franz Brentano, was professor and
was building up a cadre of outstanding young philosophers. In 1912 he
obtained his doctorate with a dissertation Przyczynek do analizy
zdań egzystencjalnych [Contributions to the Analysis of
Existential Propositions]. This was published the previous year in the
leading Polish-language philosophy journal Przegląd
Filozoficzny. Several further publications followed until the
outbreak of war, which Leśniewski spent in Moscow teaching
mathematics at Polish schools. During this period he developed what
later became known as Mereology. After the Bolshevik Revolution in
Russia, Leśniewski left Russia for Poland. During the
1919–21 Polish–Bolshevik War he worked as a codebreaker
for the Polish General Staff's Cipher Bureau, helping Poland to
survive Russia's attempt to reconquer it. He tried unsuccessfully to
obtain his habilitation in Lwów, where he was blocked, getting
it instead in the University of Warsaw, where in 1919 he became
(Extraordinary—Associate) Professor of the Foundations of
Mathematics, a position especially created for him. From then until
his final illness Leśniewski lectured regularly on logical and
mathematical topics, and built up his logical systems. Though his
logic never became widely accepted, even in Poland, with Jan
Łukasiewicz he was respected as one of the co-founders of the
Warsaw School of Logic. Leśniewski's only doctoral student,
Alfred Tarski (Leśniewski used to boast of having 100% geniuses
as doctoral students) joined them to form the world's pre-eminent
centre of mathematical logic in the interwar years. Leśniewski
was made a full professor in 1936. Always a smoker, he contracted
thyroid cancer and died on 13 May 1939 at the age of 53. His papers,
entrusted to his student Bolesław Sobociński, included
unfinished works on logical antinomies and on many-valued logic. All
these papers were destroyed during the Warsaw Rising of 1944.

Leśniewski's early writings are all papers on topics in
the philosophy of logic and language, concerned with such issues as
truth, denotation and connotation, the status of the laws of logic, and
Russell's Antinomy. The primary influences in this period were
John Stuart Mill, Anton Marty, and Edmund Husserl: he himself described
the works as grammatical in nature. Although Leśniewski later
repudiated his early works, they contain many seeds of his later
interests, attitudes, and working practices. From the beginning there
is an obsessive rigor in developing logical principles, including an
instinctively clear and consistent marking of the use/mention
distinction, that is, distinguishing talk in a language from talk about
a language. Leśniewski's early works are mainly reactions to
work by others, whether Brentano and Cornelius on existential
propositions, Łukasiewicz on the Principles of Contradiction and
Excluded Middle, Twardowski on universals, or Kotarbiński on the
timelessness of truth.

The turning point in this period came when in 1911 Leśniewski read
Łukasiewicz's groundbreaking 1910 monograph O zasadzie
sprzeczności u Arystotelesa [On the Principle of
Contradiction in Aristotle], a radical rethink of the status of the
principle of contradiction in the light of modern logic. That work
contained an appendix in which there was a short exposition of modern
symbolic logic, in the notation of Couturat, and in Chapter 18 a brief
discussion of Russell's antinomy of the set of sets which are not
members of themselves. Leśniewski initially thought the Russell
Antinomy was easy to fix—and while trying to fix it missed a
once-a-day train when changing stations in Russia. He spent the next
years perfecting his solution, and indeed the rest of his life was
devoted to providing a rigorous, antinomy-free foundation for
mathematics which avoided both the sloppiness of Principia and
what he considered the fiction of standard set theory.

According to Leśniewski's dissertation and first paper, all
positive existential propositions are analytic. This sounds absurd, but
Leśniewski has reasons. Like Mill, he distinguishes between a
term's denotation, which is the object or objects it stands for,
and its connotation, which is the property or properties it ascribes to
something. A proposition is defined to be analytic if it is (1) in
positive subject–predicate form, and (2) contains no predicates
connoting properties not connoted by the subject. A proposition is
synthetic if it is (1) positive and (2) contains some predicate
connoting a property not connoted by the subject. Contrary to Mill, who
said that ‘being’ connotes the property of existing,
Leśniewski thinks the predicate ‘exist’ does not
connote any properties. A fortiori, ‘exist’ used as a
predicate connotes no property not contained in the subject. Negative
existential propositions, which have the form ‘X is/are
non-existing’ are synthetic, except for such whose subject
connotes non-existence, e.g. ‘A square circle does not
exist’ does not, so it is synthetic, but ‘All non-existing
objects are non-existing’ is analytic. All negative existential
propositions are contradictory, because every subject connotes
something of the form ‘being which possesses properties
A, B, C, D, etc.’ Analytic
negative existentials have contradictory subjects. So instead of saying
‘X exists’ we should, to express the proposition
we want to be synthetic and contingent, rather say ‘Some being is
the object X’.

Leśniewski saves appearances, which are that some existential
propositions are true and others are false, by proposing a normative
schema that propositions should fulfill to be proper expressions of the
thought intended. Every proposition is to represent the possessing by
the object(s) denoted by the subject of the properties connoted by the
predicate. Sentences which fail to conform to this norm are improper or
inadequate, and should be replaced by proper alternatives. For example
‘People exist’, which is true but is intended to be
synthetic, should be expressed as ‘Some beings are people’,
while ‘A square circle does not exist’ should be expressed
by ‘No being is a square circle’. Here is a table of some
of the equivalences:

Inadequate Expression

Adequate Alternative

Only objects A exist

All beings are objects A

Objects A exist

Some beings are objects A

Object A exists

One (a certain) being is object A

Object A does not exist
Objects A do not exist

No being is a/the object A

In “An Attempt at a Proof of the Ontological Principle of
Contradiction” (1912) Leśniewski argues that ‘No
object is able to be both B and not-B’ is true
but disagrees with the equivalence (now generally accepted) that
‘No A is B’ is equivalent to ‘If
something is A then it is not B’. The article
is written in criticism of some ideas of Łukasiewicz and it
afforded the occasion for the first personal meeting of the two later
colleagues. Łukasiewicz described the meeting decades later in a
poignant diary entry for 9 May 1949:

Yesterday was the feast of St. Stanislaus, Bishop. This was
Leśniewski's name day. On his last one he already lay in the
same hospital in which five days later he was to die. I met
Leśniewski in Lwów in 1912. I lived then with my uncle in
10, Chmielowski Street. One afternoon someone rang at the entrance
door. I opened the door and I saw a young man with a light, pointed
beard, a hat with a wide brim and a big black cockade instead of a tie.
The young man bowed and asked politely: “Does Professor
Łukasiewicz live here?”. I replied that it was so.
“Are you Professor Łukasiewicz?” asked the stranger. I
replied that it was so. “I am Leśniewski, and I have come to
show you the proofs of an article I have written against you.” I
invited the man into my room. It turned out that Leśniewski was
publishing in Przegląd Filozoficzny an article containing
criticism of some my views in The Principle of Contradiction in
Aristotle. This criticism was written with such scientific
exactness that I could not find any points which I could take up with
him. I remember that when, after hours of discussion, Leśniewski
parted from me, I went out as usual to the Kawiarnia Szkocka [Scottish
Café, a favorite meeting-place for academics in Lwów],
and I declared to my colleagues waiting there that I would have to give
up my logical interests. A firm had sprung up whose competition I was
not able to face. [Translated by Arianna Betti and Owen LeBlanc;
see Polish Philosophy Page: Stanislaw Lesniewski, in the
Other Internet Resources section below]

In “The Critique of the Logical Principle of the Excluded
Middle” (1913) Leśniewski distinguishes between an
ontological principle – Every object is either A or not
A – and a logical principle – Of two
contradictory propositions, at least one must be true. He rejects the
latter because he takes ‘Every centaur has a tail’ and
‘Some centaur does not have a tail’ to be contradictories,
but both to be false because the subject-term ‘centaur’ is
empty. This clearly depends on a reading of the universal with
existential import. If instead for the universal we take ‘Any
centaur has a tail’, which is true if there are no centaurs,
then we do get a contradictory pair with opposite truth-values.

In the course of this paper Leśniewski argues against
Twardowski's theory of general objects and Meinong's theory
of impossible objects, as well as offering solutions to the paradoxes
of Grelling and the Liar. The paper is the richest of his early
“logico-grammatical” works. The argument against Twardowski
goes as follows. Call a general object one which has all and
only the properties shared by some group of objects. For example the
general horse has all and only the properties shared by all horses: it
is neither black nor brown nor white, neither male nor female, neither
young nor old, but it is indisputably equine, mammalian, and born of
two equine parents. Leśniewski now reduces the definition to
absurdity. Of any group of at least two objects, there will be some
property that some of them have that others do not have. For example,
some horses are black and others are not black. So the general horse is
not black, because some horses are not black; but nor is it not black,
because some horses are not not black. Hence the general horse is
neither black nor not black, which is a contradiction. Hence there are
no general objects for groups of more than one thing. Leśniewski
liked this argument well enough to retain his allegiance to it even
while repudiating all other parts of his early work; he rejected
universals consistently thereafter. But the argument does not affect
those theories of universals which are happy to accept that universals
may have properties their instances do not, for example being
instantiated, or repeated.

In “Is truth only eternal or both eternal and without a
beginning?” (1913) he argues for timeless bivalence against
Kotarbiński's view that future contingent propositions lack
a determinate truth-value. This paper convinced Kotarbiński he had
been in error. The exchange is notable for showing that the discussion
of the logical status of future contingent propositions, which inspired
Łukasiewicz to invent many-valued logic a few years later, was
already under way in Lwów before the First World War.

In “Is the class of classes not subordinate to themselves
subordinate to itself?” (1914) Leśniewski offers his first
published analysis of Russell's Paradox, claiming that ‘the
class of As’ refers to the unique mereological sum of
As, so that since every object is subordinate to itself, no
class of objects is not subordinate to itself, and Russell's
Paradox fails to arise. The non-standard term ‘subordinate
to’ which he took from Łukasiewicz, is defined by
Leśniewski as follows. An object P is subordinate to a
class K if and only if, for some a, K is a class of
as (a mereological sum of as, as we shall see), and
P is an a. Let P be a hemispherical part of
a sphere Q. The class of all hemispheres of Q is, on
Leśniewski's understanding, simply Q itself, so
P is subordinate to Q. In fact, according to this
view, any object is a class, and subordinate to itself. Leśniewski
retained this “concrete” understanding of classes
throughout his life, claiming that it conformed to Cantor's own
statements. That other set theories gave a different account of things
was in Leśniewski's view their problem, not his.

During World War I, when Leśniewski lived in Moscow, he completed
“Foundations of a General Theory of Sets I” (1916). Despite
the use of the term ‘mnogość’ (set),
this was a first rigorous deductive presentation of the theory of
parts, wholes and concrete collections. Leśniewski later dropped
the term ‘mnogość’ and instead invented
the term ‘Mereology’, meaning ‘theory of
parts’, an irregular coinage from the Greek
μερος, part, in order to
differentiate his view from what he ironically called
“official” set theory. In this paper the language is a
highly regimented Polish supplemented with variables, because at the
time Leśniewski did not trust symbolic methods, having found the
use/mention confusions in Whitehead and Russell's Principia
Mathematica a barrier to understanding, and assuming any symbolic
logic had to be like that. Only later did Leśniewski discover the
much tidier work of Frege, which he thereafter held up as the foremost
example of symbolic logic to that date. Leśniewski was one of the
earliest logicians to appreciate the virtues of Frege's logical
work as largely independent of its inconsistency. In later years,
Leśniewski proved that Frege's “way out” of the
paradox, which involved a restriction of the fatal abstraction
principle Basic Law V, had the unacceptable consequence that there
could not be more than one object. This work was not published, but was
reconstructed after World War II by Sobociński (1949).

It is important for understanding Leśniewski's development and
his attitude to logical systems as endowed with meaning from the
start, and not meaningless formal games, to know that he developed in
his systems chronologically in the reverse order from their order of
logical priority. Logically, protothetic precedes ontology, and
ontology precedes mereoology, but he worked them out starting from
mereology and working through ontology to finish with
protothetic. Becoming frustrated with the inexactnesses and foibles of
natural language as a medium for working, in 1920 Leśniewski let
himself be persuaded by Leon Chwistek that it would be advantageous to
overcome his distaste for symbolism and formulate his logical thoughts
using symbols. Mereology having been rather well formulated already in
1916, the transition to symbolism was straightforward there. The fact
that axiomatization came first, and symbolization of the already
axiomatized theory followed subsequently, gave Leśniewski a
strong reason to dissociate the use of formal methods from
formalism, according to which formulas are without
interpretation. Leśniewski's formulas had an intended
interpretation from the start.

The original (1916) formulation of Mereology, not yet so called, and
not yet formalized with special symbols, had the primitive concept
‘part’, four axioms and three definitions at its basis.
They are:

Axiom I. If object A is part of object B, then
B is not part of A

Axiom II. If object A is part of object B, and
object B is part of object C, then A is part
of C.

These specify respectively the asymmetry and transitivity of the
part relation.

Definition I. The expression ‘ingredient of object
A’ is used to denote A, and every part of
A.

That is: Object B is an ingredient of object A if
and only if either B is A or B is part of
A. Nowadays the term ‘ingredient’ is often simply
rendered as ‘part’ and Leśniewski's
‘part’ is called ‘proper part’.

Definition II. The expression ‘set
[mnogość] of objects m’ is used to
denote every object A such that if B is any
ingredient of A, then some ingredient of B is an
ingredient of some m, and this m is an ingredient of
A.

That is: A is a set of m if and only if every
ingredient of A has a common ingredient with (mereologically
overlaps) some m, and this m is part of A.
Intuitively, a set of m is what we would now call a
mereological sum, consisting of one or more m, but not
necessarily all of the m.

Definition III. The expressions ‘set of all objects
m’ and ‘class [klasa] of objects
m’ are used to denote every object A such that
(i) every m is an ingredient of A, and (ii) if
B is an ingredient of A, then some ingredient of
B is an ingredient of some m.

That is: a class of m is a set of all the m. The
remaining two axioms state that such classes exist and are unique:

Axiom III. If some object is (an) m, then some object is a
class of objects m.

Axiom IV. If A is a class of objects m, and
B is a class of objects m, then A is
B.

With these axioms we may speak of ‘the class of
m’ whenever there is at least one m.

In terms of these basic principles, Leśniewski proves a number of
theorems and defines several important mereological concepts, such as
overlapping (having an ingredient in common) and being exterior to
(having no ingredient in common). It is characteristic of this essay
that Leśniewski is keen to appropriate the terminology of set
theory for his own purposes. Further terminology includes
‘element’, which is defined as follows:

Definition IV. The term ‘element of object A’
is used to denote any object B which, for some meaning of the
expression ‘x’, is such that (i) A is the
class of objects x, and (ii) B is (an)
x.

It is swiftly proved that the ingredients of A and the
elements of A are the same. The definition illustrates how
Leśniewski formulated the idea of quantification in his early
semi-prose work: instead of “for some x” he says
“for some meaning of the expression ‘x’
”. When showing all ingredients of an object are its elements,
Leśniewski instantiates the bound variable by saying “Using
the expression ‘x’ with the meaning of the
expression ‘ingredient of the object A’
…”. We will return to this below when discussing
Leśniewski's understanding of the quantifiers.

By Leśniewski's own later and very exacting standards, this
first formulation of Mereology is methodologically imperfect, because
it intersperses axioms and definitions. A cleaner formulation would
express all axioms in terms of the mereological primitive (here
‘part’) only. This would be possible in this case simply by
substituting the definitions of defined terms in Axioms III and IV, but
it would not be especially enlightening. It also results in an axiom
system that can be substantially simplified, both by reducing the
number of axioms and by simplifying and shortening them. These
desiderata (fewer, shorter, and more perspicuous axioms) often
pull in different directions.

The language of Mereology in 1916 used many expressions besides the
specifically mereological primitive ‘part’: in addition to
nominal variables, there are expressions forming sentences from nominal
variables, as in ‘A is (a) b’,
‘A is B’, ‘every a is a
b’, ‘some a is a b, ‘no
a is a b’, as well as words such as
‘object’ and ‘exist’. Complex names also occur
in sentences, as the expressions ‘part of A’ and
‘ingredient of A’ in ‘every part of
A is an ingredient of A’. Leśniewski had
hitherto taken such logical bits of language for granted, but now he
needed to formalize them. He wanted a logical calculus of names and the
expressions involving them. There were precedents in traditional
syllogistic, and more especially in the algebra of logic of Ernst
Schröder, which Leśniewski looked at in coming up with his
own system, based as in the case of Mereology on his intuitive
understanding of the relevant expressions as carefully used in ordinary
language. At first he gathered propositions he was sure were true, such
as ‘If A is b, then A is
A’. We know about this one because it is mentioned,
presumably scornfully, in a diary entry by Twardowski for 1 July 1919,
as being the first axiom of the new system that Leśniewski was
working on at the time. Leśniewski has left us a graphic
description of his method of working at this crucial and fluid time of
his development [Collected Works 366-9]:

While using colloquial language in scientific work and attempting to
control its ‘logic’, I endeavoured to somehow rationalize
the way in which I was using in colloquial language various types of
propositions passed down to us by ‘traditional logic’.
While relying on ‘linguistic instinct’ and the often
non-uniform tradition of ‘traditional logic’, I attempted
to devise a consistent method of working with propositions that were
‘singular’, ‘particular’,
‘general’, ‘existential’ etc. The results of my
efforts were useful and I continued my efforts in applying to the
‘symbolism’ the equivalents of various types of
propositions, after the change to the ‘symbolic’ way of
writing.

While working in this way, and attempting to define some expressions
in terms of others, Leśniewski came to focus on singular
propositions of the form ‘A is (a) b’,
which he henceforth wrote as ‘A ε
b’, borrowing from Peano the lower-case epsilon, the
first letter of the Greek ‘εστι’,
is. It was this connection with meanings of ‘is’
that prompted Leśniewski to name this system
‘Ontology’. He thought it would be possible to base it on
singular inclusions like ‘A ε b’
alone, in addition to concepts taken from the logic of connectives and
quantifiers. The key thought was that the following should be true

A is a if and only if (every A is
a, and at most one object is A)

This requires the two expressions ‘every A is
a’ and ‘at most one object is A’ to
be defined. The first can be helped along by

every a is b if and only if (some object is
a, and for any X, if X is a, then
X is b)

which requires ‘some’ and ‘object’ to be
defined: they can be helped further by

some a is b if and only if for some X,
X is a and X is b

if A is b, then A is an object

while the second can be helped by

at most one object is a if and only if for any A and
B, if A is a and B is a,
then A is the same object as B

and finally we have

A is the same object as B if and only if (A
is B and B is A).

Putting these together, and inspired by Russell's theory of
definite descriptions, Leśniewski arrived in 1920 at a single
axiom based on the single primitive ‘is’:

A is a if and only if ((for some B,
B is A) and (for any B and C, if
B is A and C is A, then B
is C), and (for any B, if B is A
then B is a))

The story has it that Leśniewski discovered this axiom, his
efforts fortified by eating bars of chocolate, while sitting on a bench
in Warsaw's Saxon Garden. Symbolically, using a somewhat more
modern symbolism than Leśniewski's, but borrowing his upper
corners for marking quantifier scope, the axiom becomes

Several things are notable about this axiom and its formulation. It is
in the form of a universally quantified equivalence, with the
right-hand side as it were explicating the left-hand side, so that it
is a kind of implicit definition of the primitive
‘ε’. The right-hand side, taking its cue from
Russell, says that there is at least one A (first conjunct),
there is at most one A (second conjunct), and any A
is an a (third conjunct). The use of capital letter variables
is an informal aid to perspicuity: they mark the positions in
sentences (especially before ‘ε’) where a variable
can only yield a truth of its immediate context if it is
singular. Where small Italic variables are used, there is no such
presumption of singularity. In principle only one typeface of
variables is needed. On the basis of his axiom and several rules of
inference, Leśniewski developed a powerful system of general
logic comparable in strength to a simple type theory: he writes in
1929, “In 1921 I developed my ‘theory of types’
[…] It was something like Whitehead's and Russell's theory of
types, which I had generalised and simplified in a certain way”
(Collected Works, 421).

Both Mereology and Ontology presuppose a deeper logical layer,
comprising a propositional logic of connectives like ‘if’,
‘not’ and ‘and’, and the as yet unexplicated
logic of the quantifiers ‘for all’ (‘∀’)
and ‘for some’ (‘∃’). Having settled
Ontology on an axiomatic basis, Leśniewski turned to the
axiomatization of this. He originally spoke of the “theory of
deduction”, which was the name Whitehead and Russell used for the
propositional calculus, but because they did not introduce a logic of
quantifiers until later, he coined the term ‘Protothetic’,
from the Greek for ‘first theses’. It is characteristic of
Leśniewski's logic that he introduces quantifiers in the
most basic part of his logic, even before names are introduced. This is
unlike most modern theories, which introduce quantifiers only when
names and predicates are brought in. It leads to questions about the
nature of quantification in Leśniewski which we will revisit
below.

Leśniewski's preference for an axiom system, based in part
on the success of Ontology, and also on considerations about the nature
of definition, was to base a logical system on the single connective of
material equivalence, together with the universal quantifier. He was
held up for some time in doing this for Protothetic by his inability to
see how to eliminate the connective of conjunction in terms of
equivalence. Given quantification and equivalence, negation is easy to
define, in a way Russell once suggested to Frege:

Def.~ :

∀p⌈~p ↔
(p ↔ ∀r⌈r⌉)⌉

The solution was found for Leśniewski by his 21-year-old
doctoral student Alfred Teitelbaum, later renowned under his adopted
name as Alfred Tarski. It consisted in quantifying not just sentences
but sentential functions or connectives:

Def.∧:

∀pq⌈p ∧ q ↔
∀f⌈p ↔ (f(p)
↔ f(q))⌉⌉

in this case, quantifying one-place connectives. Assuming there are just four of
these connectives, assertion, negation, Verum (tautology) and
Falsum (contradiction), it is straightforward to show that the
right-hand side is equivalent to the conjunction of p and
q. Tarski's doctoral dissertation centres around this
result.

As to axiomatization, Leśniewski knew that the pure theory of
equivalence could be based on two axioms stating skew-transitivity and
associativity:

P1

((p ↔ r) ↔ (q ↔ p))
↔ (r ↔ q)

P2

(p ↔ (q ↔ r)) ↔ ((p
↔ q) ↔ r)

Pure equivalential calulus has the quaint property, shown by
Leśniewski, that a formula is a theorem if and only if every
propositional variable in it occurs an even number of times. After
universally quantifying these axioms, a further axiom was added to
introduce propositional functions, in this case two-placed ones

Once again Leśniewski and his students sought after shorter and
more perspicuous formulations, or ones consisting of a single axiom,
though the latter tended to be neither short nor perspicuous.

With Protothetic launched, Leśniewski could now look back on his
system of foundations and see that it consisted of a hierarchy of three
systems, developed in reverse order: Protothetic, introducing
connectives, quantifiers and higher functions; Ontology, introducing
the new category of names, with the new primitive ‘is’, and
Mereology, based on a primitive mereological functor such as
‘part of’ or ‘ingredient of’, but introducing
no new categories of expressions not already foreseen in Ontology.

One of Leśniewski's most lasting contributions to the metatheory
of logic is his theory of semantic categories. This replaced the
theory of simple types which he developed in 1921, about which he
wrote, “[E]ven as I as I was constructing my theory of types, I
considered it to be only an inadequate stop-gap […] In 1922 I
outlined a concept of semantical categories as a replacement for the
hierarchy of types, which is quite unintuitive to me”
(Collected Works, 421). In type theory, expressions belonging
to different logical types cannot be substituted for one another
without turning grammatical or well-formed expressions into
ungrammatical or ill-formed ones. Only well-formed or grammatical
expressions may have a sense or meaning. The theory was developed by
Bertrand Russell as a way of blocking set-theoretic paradoxes, though
there were anticipations in the work of Ernst Schröder and
Gottlob Frege. In type theory it is usually assumed that variables at
each type range over a domain of entities specific to that type, and
that all such domains are mutually disjoint. In Frege for instance
the domains were objects and functions of various levels, while in
Russell they are usually taken to be a hierarchy of propositional
functions. This presumptively platonistic and ontologically
inflationary position was naturally uncongenial to the nominalist
Leśniewski, and he readjusted the notion of category, turning
categories from classes of entities (with their attendant expressions)
to classes of expressions alone. His inspiration was in part the
traditional syntactic theory of different parts of speech, and in part
Husserl's theory of Bedeutungskategorien (meaning categories)
from the latter's Logical Investigations. Where Husserl's
categories were of abstract meanings, Leśniewski, ever the
nominalist, substituted categories of (concrete) expressions. Although
like later writers he could have called the classes of expressions
‘syntactic categories’, he deliberately chose the
expression ‘semantic categories’ in order to emphasize
that the expressions combined grammatically are all meaningful, unlike
the meaningless marks proposed by formalist writers of the Hilbert
School.

Leśniewski himself never gave an explicit formulation of the
theory of semantic categories, being content to work with them in
practice. The first formulation was by his contemporary Kazimierz
Ajdukiewicz, in the 1935 article ‘Syntactic Connection’.
Ajdukiewicz's essay became the fountainhead of the subsequent
subdiscipline of categorial grammar. Modifying Ajdukiewicz's
notation, we may explain semantic categories as Leśniewski used
them. Leśniewski thought that the theory applies only to
his logical systems, not to ordinary language,
about which he became rather sceptical as to its ability
to be unambiguously precise. Subsequently it has been shown that
categorial grammar can be applied quite successfully to the syntax of
natural languages.

There are in Leśniewski two basic categories: sentence (S) and
name (N). In Protothetic, only the former is used: Ontology and
Mereology add the latter. The distinction between sentences and names
is ultimate: what we can say about sentences is that they are there to
say things that are true or false (obviously from a logical point of
view we are overlooking questions and commands), whereas names are
there to denote things. Leśniewski, following tradition, allows
names to denote several things, or one thing, or indeed no things as
all. So ‘Istanbul’ denotes one thing, namely the Turkish
city, ‘city’ denotes many things, namely all cities, and
‘unicorn’ denotes nothing at all. In
Leśniewski's mature work Mill's notion of connotation
is dropped, along with the concept of property which it employs, so the
sole logical function of names is denoting. Again following tradition
rather than the modern approach of Frege and Russell, Leśniewski
makes no syntactic distinction between general terms or common
nouns on the one hand and singular terms or proper names on the other.
This is often said to be because his native Polish lacks definite and
indefinite articles, which make the distinction more grammatically
obvious, but this conjecture is nonsense because Leśniewski spoke
and wrote fluent German, and German is awash with articles. It seems much more
likely that Leśniewski deliberately chose to follow the
traditional rather than the modern way because he thought it both more
expressively powerful and closer to natural language.

Since language does not consist solely of unarticulated sentences or
names, there are expressions of other categories, which combine in a
rule-governed way with one another to produce further expressions,
ultimately sentences. In the regimented environment of
Leśniewski's logical languages this always takes place in
the following way: a combining expression, which we may call a
functor, precedes a left parenthesis of some kind, which is
then followed by a sequence of one or more argument
expressions, followed by a right parenthesis symmetric to the other
one, which terminates the complex. The general schema is then

Now let us provide a notation, inspired by Ajdukiewicz, for the
category of a functor such as ‘F’. If the category
of ‘a1’ is α1, the
category of ‘an’ is
αn, and the category of the whole expression
‘F(a1…
an)’ is β, then the category of the functor
expression ‘F’ we write as

β〈α1…αn〉

which indicates the category of the output to the left and the
category of its inputs in order within the angled brackets. Call this
the categorial index of the expression. Thus the category of
sentential negation is
S〈S〉,
that of conjunction is
S〈SS〉,
while that of the epsilon functor ‘ε’ is
S〈NN〉,
since it builds a sentence using two names as arguments.

What Ajdukiewicz showed in his 1935 paper was that we can develop a
calculus of grammatical combination using such a notation: we take a
putatively well-formed expression, rearrange it if necessary into
functor-first order, then see if we can “multiply out”
arguments and functors to arrive at a single categorial index. If we
can, the complex expression is grammatical, well-formed, or
syntactically connected. For example
‘ε{Aa}’ is syntactically connected as
follows: writing the category of an expression e as
|e| we have

|ε| =
S〈NN〉,
|A| = |a| = N, so |ε{Aa}| =
S〈NN〉
× (N × N) = S

as we would expect. Ajdukiewicz used a “quotient”
notation instead of our angled brackets; this makes the idea of
“multiplying out” more graphic, but becomes cumbersome for
complex cases.

There may be functors whose arguments are functors: for example a
conjunction holding between two binary predicates has a category
S〈S〈NN〉S〈NN〉〉.
There may also be what are called many-link functors, which
are functors whose values are functors. For example, the English
morpheme ‘–ly’ converts an adjective, category
N〈N〉,
into an adverb, for example of category
S〈N〉〈S〈N〉〉,
so ‘–ly’ has effective category
S〈N〉〈S〈N〉〉〈N〈N〉〉.
In some subsequent categorial grammars, transitive verbs are quite
plausibly taken to have the many-link category
S〈N〉〈N〉
rather than the binary predicate category
S〈NN〉.
This move is indeed a standard trick in logic, first introduced by
Moses Schönfinkel in 1924, for dispensing with many-placed
functors in favor of many-link but one-placed functors. Had he known
of this, Leśniewski would no doubt have disapproved. While there
is no loss of logical power in eliminating many-placed functors, the
move is unnatural, and Leśniewski would not have countenanced
defining many-placed functors as many-link ones in disguise,
as we find in Church for instance.

With the two basic categories of sentence and name, every category and
expression, no matter how complex, will have a categorial index
beginning either with ‘S’, so ultimately leading to the
formation of a sentence, or ‘N’, so ultimately leading to
the formation of a name. Following a useful terminological suggestion
of Eugene Luschei, we may call the former propositive
categories and expressions, the latter nominative categories
and expressions (Luschei 1962, 169).

Note that the parentheses in Leśniewski's notation do not
themselves have a category: they are syncategorematic. Their function
is twofold: to mark the beginnings and ends of argument strings, and to
help indicate the semantic category of the functors. Thus
Leśniewski used round parentheses for functors yielding sentences
from sentences, i.e. connectives, and braces for functors yielding
sentences from names, i.e. predicates. In principle an unlimited number
of shapes of parentheses might be required, and indeed in some works of
students of Sobociński there are several dozens of different
shapes. Leśniewski gave parentheses this second role because he
wished to remain very flexible about the forms of expression used for
functors, even allowing the same shape to be used in
“analogous” functors, for example 3-placed conjunction, or
“higher-order” epsilons and equivalences, or other logical
constants. Obviously this is a contingent feature of his notation:
other conventions will do just as well.

More important for metalogical purposes is the fact that the universal
quantifier too is syncategorematic in Leśniewski. This is marked
symbolically in that the lower corners which he used to symbolize the
universal quantifier are no more than a container for the variables,
but more crucially, in that any finite string of different variables
may occur in such a quantifier, no matter how assorted their
categories. There are some advantages in this
flexibility. Leśniewski does not need to give rules for many
different kinds of universal quantifier, but gives rules for the one
sort in one go. But there are some disadvantages too. In his
“official” notation for his logics, Leśniewski has
only the universal quantifier, and does not define the particular
quantifier or any others in the standard way. This was also Frege's
practice, but whereas in Frege's case the parsimony seems to have been
self-inflicted, in Leśniewski's case there is a systematic
reason. Leśniewski was scrupulous about giving exact rules for
admitting new expressions by definition. He gave such rules for
Protothetic, and extended them to Ontology. These rules govern only
basic and functor category expressions. Quantifiers, being variable
binders, are neither basic nor functor expressions, but
Leśniewski was unable to come up with acceptable canons of
definition for such variable binders. He would have liked to have done
so, and indeed offered students any degree they wanted, from Master's
to
Habilitation, if they could formulate adequate rules, but no
one could. Therefore in the “official” system the
universal quantifier remained as a syncategorematic expression, but
still entered into legal combinations, which meant that the syntax of
his systems was not fully captured by categorial grammar. Such a
limitation became apparent to Ajdukiewicz also, who made an
unsuccessful attempt to rectify it. Ajdukiewicz noted perceptively
that a language containing Russell's circumflex abstraction operator,
which Alonzo Church had notated using the Greek lambda, would be able
to express any operator as a combination of an abstraction operator with a
functor. Church employed this method to considerable advantage in his
logic, but this work came too late to help Leśniewski. In any
case, in Church it simply pushed the problem of syncategorematicity
over to the lambda operator.

The status of definitions in logic and outside it has been
surprisingly controversial. Probably the standard view among logicians
is that of Russell, according to which definitions are mere
abbreviations whose role is to render complex propositions more easily
surveyable by feeble humans. According to this view, when a logician
defines, say, conjunction in terms of implication and negation, as

p ∧ q =df
~(p → ~q)

this is to be understood as merely a shorter expression substituting
for the longer one. In this case the abbreviatory value is negligible,
but in the case of some long expressions, such as that for the
ancestral of a relation, or in very many areas of mathematics, it may
be considerable. Russell expressed this view somewhat
over-dramatically by calling a definition an expression of the
author's will. On this view, a definition cannot be true or false; it
may be appropriate or helpful, or not, and there are certain
proprieties to be observed, such as not having dangling variables in
either definiens or definiendum, and not trying to define an
expression in terms of itself, but beyond this, the methodological
requirements on abbreviation are minimal. In many modern logical
systems, definitions are said to be confined to the metalanguage only,
and not to appear in the object language at all.

There is nothing wrong with this view, but it is emphatically not
that of Leśniewski. According to him, a definition introduces a
new expression into the object language. Again, the proprieties have
to be respected, and getting these right turns out to be a tricky
matter. Also, it is up to the author of a logical system as to which
definitions he or she chooses to introduce. But there is an important
difference. Since definitions add new expressions to the object
language, they add expressions in places where they can be quantified,
and so can enhance the system's expressive power. Leśniewski
admits this option: according to him it is the logician's job to keep
adding to the system. If in the process, new things are provable that
were not provable before, so much the better, provided the proprieties
are respected. If we can prove a theorem not containing a defined
expression after a definition has been introduced that could
not be proved before, the definition is said to be
creative. Most logicians decry creative definitions:
Leśniewski embraced them. Leśniewski contended that the
symbol ‘=Def.’ used by most logicians actually sneaks an
unrecognized primitive into their work. He thought this of Whitehead
and Russell, and it is why he wanted to formulate Protothetic based on
equivalence alone, since then the same connective is primitive that is
used for definitions. In his view, definitions are object-language
equivalences and should be recognized as such. In retrospect we can
see that this view was too extreme. There is room for abbreviatory
definitions, and indeed Leśniewski “inoffically” used
one himself, that of the particular quantifier. What Leśniewski
calls ‘definitions’ might better be called
‘definitional axioms’. That these can be added as we go
along rather than collected together at the beginning of a logic is
simply a feature of his way of doing logic.

Perhaps the most important creative definition used by
Leśniewski is one from Ontology, due to Tarski in 1921. Is is for
a functor * of category
N〈N〉:

∀AB⌈A ε *(B) ↔
∃c⌈A ε c ∧
B ε A⌉⌉

where ‘A ε *(B)’ can be read as
‘A is the unique B’. It allowed the long
axiom of Ontology to be replaced by the short one (see below): without
it, the replacement was not possible.

We noted that Leśniewski, persuaded by his argument against
Twardowski's general objects, was a nominalist. It was no doubt part
of his antipathy towards set theory. His friend Kotarbiński
formulated a very extreme nominalism, called variously reism,
pansomatism, and concretism, according to which the only things that
exist are material bodies. Leśniewski would not go this far,
because he did not see how after-images and dreamt things, which he
thought existed, could be material bodies. But in elaborating his
logic he emphatically rejects anything which is not concrete,
individual, located in space and time. This goes for the logical
systems themselves. His view, which would nowadays be called
inscriptionalism, is that logical systems are actual
collections of concrete marks, whether printed on paper in books and
journals, or in handwritten notes, or more ephemerally as spoken
words, chalk marks on blackboards, or (nowadays) patterns on computer
screens. Leaving aside the non-trivial metaphysical question of what
can in principle count as a logical inscription, we must consider what
effect this stance had on his attitude to logic and logic systems.

The effect is far-reaching and radical. If a logical system is a
concrete complex of signs, then it cannot be infinite. Also, to
conform to logical practice, it has to be admitted that logical
systems change over time. Ideally, they change by being added to, as
new theorems are proved. In practice, they may degrade or be wholly
destroyed, which was indeed the sad fate of Leśniewski's own
personally written systems in November 1944. If a logical system is
published in a book or journal, and there are several copies, then
there are as many such systems as there are copies. Assuming the
copies are all faithful, then each copy is typographically exactly
like all the others. They are, in Leśniewski's parlance, all
equiform. In practice of course, equiformity is not quite
exact, even in printed works, but the minute variations are
insignificant, and in any case we wish to recognize handwritten
manuscripts and other variants to be equiform for logical purposes
with systems which are physically somewhat different. Again the
metaphysical fine detail is less important that the fact that we get
by most of the time with constrainedly inexact similarity.

Almost all metalogical ways of dealing with logical systems are
platonistic. They assume that the simple and complex expressions are
abstract types, that they are infinite in number, that an axiom system
has infinitely many logical theorems, and so on. Leśniewski can
admit none of this. So he has to find a way to deal with logical
systems as something organic, growing and changing with time. He does
so by employing a complex system of metalogical definitions, which he
calls ‘terminological explanations’, and inference rules,
which he calls ‘directives’. In effect Leśniewski
gives a schematic grammatical and logical framework for an
indefinitely extensible language, decades in advance of the advent of
formal descriptions of computer programming languages, which are the
nearest equivalent elsewhere.

It is hard to give a flavor of the terminological explanations (TEs)
in a short space: their complexity and cumulative effect has to
appreciated at first hand. The most thorough account is in the TEs for
Protothetic in Grundzüge, Section 11, and there is also
a compressed account of TEs for Ontology. A somewhat more manageable
set for a version of propositional calculus with definitions, based on
Łukasiewicz's bracketless notation, is given in the 1931 paper
‘Über Definitionen in der sogenannten Theorie der
Deduktion’, and based on lectures from 1930–31. There
Leśniewski gives TEs in words rather than symbolic abbreviations,
and with copious examples. Nevertheless, it is intellectually
challenging, since for all the complex TEs (metalogical definitions),
Leśniewski demanded logical independence of all the various
clauses, to be shown by suitable models. As a result, it took graduate
students three semesters to work through a set of TEs in
Leśniewski's seminar (personal information from Czesław
Lejewski) .

The TEs are a means to an end: that of formulating a system's
directives. A directive sounds like an imperative, but its
illocutionary force is subtler. Assume a logical system has been
developed up to a certain point, that is, up to a certain last written
thesis (Leśniewski's word covering axioms, theorems and
definitions). At the very least, the axiom or axioms will have been
written down. Assume for the sake of argument that the development has
been fine up to now. There is no categorical imperative to extend the
system by adding another thesis, but let us suppose the system's
author (or indeed any assistant) wishes to do so. He or she writes
down a new collection of marks. Not just anything goes however. The
marks have to be legible (obviously), unequivocally dissectible into
elementary symbols (that Leśniewski calls words),
grammatically well-formed as a sentence (no unbound variables are
allowed), and finally admissible according to the
directives. The directives lay down what may be admitted next
after a given sequence of theses. For example, the next thesis may be
an instantiation from a previous universally quantified thesis, or a
modus ponens from two previous theses, or a quantifier
distribution from a previous thesis, or a definition acceptable
according to the canons, or a thesis of extensionality. Admissibility
is always relativized to the prior sequence: the order of introduction
matters. If the new candidate thesis is admissible according to one of
the directives, it passes the test and becomes part of the system,
which can then be extended still further. Otherwise it is rejected and
the system is not extended.

Leśniewski's supreme achievement as a logician, in his own eyes,
was to formulate the metalogical requirements for extending a concrete
system by directives, relative to the previous state of the
system. Since the system is not planned or fixed in advance, the TEs
and directives have to be schematically flexible enough to accommodate
any future additions, while not being so liberal as to allow
contradictions or nonsense to arise. Finding that balance, most
especially in the rules for admissible definitions, was a considerable
feat. The definition of ‘definition’ for Protothetic run
to 18 separate complex clauses, and in Ontology, which adds a second
style of definition, another 18 clauses.

Definitions, which seem so peripheral to most logical systems, are in
fact key to the potential power of Leśniewski's logic. The axioms
of a system generally use very few semantic categories near the bottom
of the recursive hierarchy. It is by definitions that new semantic
categories are introduced into the system; once a new category is in,
variables for it may be introduced and quantified, a thesis of
extensionality formulated for it, and it can provide arguments for
further, higher-level functors. So although, unlike a typed logic
conceived platonistically, only finitely many types or categories are
actually in play in any one system, the potential to go on is
restricted only by contingent limitations.

The easiest way to get an idea of how definitions work in
Leśniewski's logic is not to look at the published work on
Mereology or Protothetic, but at the extended list of
“Definitions and Theses of Leśniewski's Ontology”
from S. Leśniewski's Lecture Notes on Logic, published
in 1988. Taken from student notes of a 1929–30 lecture
course ‘Elementary Outline of Ontology’, they comprise one
axiom, 59 definitions and 633 listed (not proved) theorems, covering
syllogistic, Boolean algebra, the notions of property(-predicate) and
higher-order property, relations, higher-order epsilons, and several
chunks of the theory of relations including converses, fields and
relative products, as known from Peirce, Schröder, Whitehead and
Russell. Not coincidentally, the sketched development includes a large
number of styles of parentheses.

Much has been written about the way in which Leśniewski
understood the quantifiers ‘for some’ (∃) and
‘for all’ (∀). There are three aspects to the
controversy: (1) how the quantifiers are to be read; (2) how they are
to be understood; and (3) the logical and philosophical significance
of the way they are understood. The controversy arose principally
because of a discrepancy between the way in which Leśniewski and
Leśniewskians take the quantifiers to mean, and the orthodox way
of understanding them, as formulated in particular by Quine.
Leśniewski's student Lejewski recounts how after moving from
Poland to Britain he was surprised to find that the local (Quinean)
understanding was very different from the one he had grown up
with. The problem is exacerbated by the fact that Leśniewski did
not produce or even envisage a semantics for his logic, considering,
as had Frege and Russell, that his system was already meaningful and
had no need of a semantics to be grafted onto it from
outside. Historically it is interesting that Quine's preoccupation
with the idea of ontological commitment and its connection to
quantification and its domain, go back to discussions he had with
Leśniewski when he visited Warsaw in 1933. Quine relates that he
and Leśniewski stayed up until late at night disputing whether
the use of higher-order variables committed Leśniewski to
platonic objects, as Quine thought, or not, as Leśniewski
thought. Obviously to a nominalist like Leśniewski the thought
that his cherished system should involve him in unwanted ontological
commitments would be most repugnant.

As to the reading of the quantifiers, while we note that in his
pre-symbolic writings Leśniewski favored expressions like
“for some meaning of the expression ‘x’
” and “for every meaning of the expression
‘x’ ”, which may give us a hint as to how
he understood quantification, he later preferred the unadorned
‘for some’ and ‘for all’, followed by the
relevant variables, and it seems sensible simply to follow this.

The meaning or interpretation of the quantifiers is a more subtle
question. It seems very likely that Leśniewski thought of the
quantifiers as a notational necessity when a domain could be infinite,
since infinite disjunctions and conjunctions were not possible. In
Protothetic, as its computational variant makes clear, there is
strictly no need for quantifiers, since each semantic category, no
matter how high in the hierarchy, has only finitely many possible
(extensional) values. But in Ontology, where there is no logical
requirement that the domain of individuals be finite, the quantifiers
are indispensable. One thing for sure is that they cannot be given an
existentially committed reading, where ‘for some’ means
‘there exists’. The reason is that the laws of
quantification and the definability in Ontology of a necessarily empty
term ‘Λ’, for which it is true that no Λ
exists, entail the truth of the quantified proposition ‘for
some a, no a exists’ (cf. T.127 of
“Definitions and Theses of Leśniewski's Ontology”
in S. Lesniewski's Lecture Notes in Logic). This would be
contradictory if ‘for some’ meant ‘there
exists’. Leśniewski always used the expression
‘particular quantifier’ rather than ‘existential
quantifier’. So the question is how the quantifiers are to be
understood if not in the standard way.

Quine took from his discussions with Leśniewski the idea that
the quantifiers were somehow substitutional. There is an element of
truth in this. A universally quantified formula
∀X…⌈—X—⌉
licenses the inference to any formula —C—
obtained by substituting any well-formed expression C of the
appropriate category in place of the bound variable X, and
likewise for all the other variables occuring bound by the same
quantifier. Dually, from —C— one may infer
∃X…⌈—X—⌉ for
suitable categories of bound variable. These are the usual quantifier
rules, liberalized to apply to any categories of variable in batches
of one or more. So for example if there is a theorem universally
quantifying nominal variables, such as
‘∀a⌈any a is an
a⌉’, we can validly infer ‘any Λ is a
Λ’ even though the name is empty. Quine assumed that the
lack of ontological commitment must entail that the quantifiers range
over expressions rather than things. This would not have been
acceptable to Leśniewski, because it would constitute use/mention
confusion, and because while it is true for the nominalist that there
are only finitely many expressions, it is not known to be true, and
may in fact be false, that there are only finitely many things.

Is there then a third way to understand the quantifiers that is
neither referential nor substitutional? Such a way has been suggested
by Guido Küng, based on the young Leśniewski's reading of
the quantifiers as “for all [some] meanings of the variable
‘x’ ”. Take this preamble to be what
Küng calls a prologue functor, which mentions
the expression ‘x’, but takes its matrix (the part
after the quantifier, within the upper corners) to be a context of
use of the variable, and takes the variables to range over
meanings, in Leśniewski's case, extensions (Küng 1977).
Again there is something right and something wrong about this. If
extensions are (as standardly understood) various kinds of sets,
nothing could be more abhorrent to Leśniewski. He laid great
store on logic
not being committed to the existence of anything, being
ontologically neutral: no existence statement is a theorem of
any of his systems. So to have found that after all he is committed, by
the back door as it were, to the hated sets, would have been a bitter
blow. The correct insight of this understanding is that neither objects
nor expressions are what quantifier expressions range over. They do
not range over extensions either: they do not range over
anything. But for each category of expression there are numerous ways
in which an expression of that category can mean. A sentence can be
true or it can be false. A name can name one thing, or several
things, all things, or no things at all. Functor expressions mean
according to the way the outputs of their combinations mean when
their inputs have certain meanings. Given a domain of individuals, in
principle all possible ways of meaning are delimited for all categories
of expression. Quantifiers tap into this potentiality. But it goes
against the nominalist grain of Leśniewski's thought to
reify all the different potential ways an expression could mean as if
these were additional objects. So when Leśniewski told Quine his
use of quantified variables in categories other than names did not
commit him to platonic objects, he was being truthful. What he was
committed to, from the moment he accepted bivalence of truth and
falsity onwards, was that expressions of the various categories may be
variously meaningful.

We have seen how Leśniewski developed his logical systems in
the struggle to provide an antinomy-free foundation for mathematics,
comparable to those of Frege or Whitehead and Russell, but without
their defects. His gradual move from a highly stylized
prose-with-variables to fully formalized systems precluded him from
considering his logic as an uninterpreted system, as did Hilbert. From
the start, he considered his systems, even the fully formalized ones,
to be comprised of constant expressions, primitive and defined, with a
fixed intended meaning, which he attempted to make clear by examples
and elucidations. He also considered all his axioms and theorems to be
true. In this he was following Frege and Russell, who likewise did not
envisage an external source as somehow conveying a meaning upon
expressions and truth upon sentences of logic.

This attitude to logic began to be overtaken by the development of
logical semantics, not least at the hands of his own former student
Tarski. The turning point came with the publication of Tarski's
essay on the concept of truth in the languages of deductive sciences.
This paper was produced in a preliminary version in 1929–30,
updated when Gödel's incompleteness results became known in
1931, and eventually published in Polish in 1933. Leśniewski was
known to be opposed to it. There are probably two reasons for this. One
is that in his metalogical apparatus Tarski avails himself of set
theory. Even though his use is not extensive, it is an incursion of set
theory into the metatheory of logic, of which Leśniewski could not
but disapprove. The other reason is probably that whereas in earlier
parts of the monograph Tarski adheres faithfully to a conception of
finite types akin to Leśniewski's theory of semantic
categories, in later parts he distances himself from this as an
unwarranted restriction and accepts the propriety of transfinite
types.

There is in fact nothing inherently anti-semantic about
Leśniewski's logical systems. They may be given more
standard formulations and considered model-theoretically (cf. Stachniak
1981). They may also be investigated metalogically in their own terms,
and without reneging on Leśniewski's scruples about abstract
entities. The fact is that few people have considered it worth the
effort of pursuing the project.

Throughout the 1920s, Leśniewski and his students worked hard
to improve the logical systems by finding single axioms, shorter
axioms, trying out new primitives, and generally seeking logical
perfection. The work, and Leśniewski's involvement in
teaching, were so intense that for several years he published nothing.
As this became something of an embarrassment, since his results were
being quoted without being in print, he resolved to postpone a fully
systematic exposition and present instead a more autobiographical
account of how the systems arose and were improved. This took the form
of two series of articles from 1927–31. One series, ‘O
podstawach matematyki’ [‘On the Foundations of
Mathematics’] appeared in the premier Polish philosophy journal
Przegląd Filozoficzny, and, eschewing mathematical
symbolism, was dedicated to an up-to-date exposition of Mereology. The
other series, ‘Grundzüge eines neuen Systems der Grundlagen
der Mathematik’ [‘Fundamentals of a New System of the
Foundations of Mathematics’] began to appear in 1929 in the
mathematical journal Fundamenta Mathematicae, and was
dedicated to Protothetic. Its 81 pages and 11 sections ended with the
promise ‘Fortsetzung folgt’, ‘Continuation
follows’, but it did not, because in the meantime Leśniewski
had fallen out with the other editors of the journal over the status of
set theory. The article did not get beyond the preliminaries of
Protothetic, setting out the history, axioms, and rules (directives)
for extending the system, and outlining a number of variants, but not
starting the deductions proper. It was not until a new logic journal
Collectanea Logica was set up in 1938 that Leśniewski was
able to continue. After a 60-page introduction to the continuation,
‘Einleitende Bemerkungen zur Fortsetzung meiner Mitteilung u.d.T.
“Grundzüge eines neuen Systems der Grundlagen der
Mathematik” ’, summarizing the earlier article and bringing
the story up to date, there followed another 83 pages comprising
Section 12, and listing twelve definitions and 422 theorems of
Protothetic, with skeletal information about how they are derived,
expressed in Leśniewski's own idiosyncratic notation for the
connectives and quantifiers. The journal did not appear in print
because of the outbreak of the Second World War: the printing plates
from which offprints had fortunately been made were destroyed in the
bombing of Warsaw in September 1939, by which time Leśniewski was
already dead.

Leśniewski's own notation for Protothetic is explained in
‘Supplementary Remark III’ of the Introductory Remarks:
one-place connectives are made up of a horizontal bar
‘–’ supplementable at either end by a vertical bar
‘|’. A bar at the right-hand end of the line signifies the
connective gives output value T(rue) for input value T, the
absence of a bar signifies the output value for input T is F(alse).
Similarly a bar at the left-hand end of the line signifies output value
T for input value F, the absence of a bar signifies output F for input
F. The four extensional one-place connectives are thus systematically
notated, with negation, the most important, written as
‘⊢’. For two-place connectives bars are added
radially as spokes to a circular central hub
‘○’. As before the presence of a bar
indicates an output value T, its absence an output value F. The top
position is for first and second inputs F, the bottom position for
first and second inputs T, the left position for first input T and
second input F, the right position for first input F and second input
T. Thus all sixteen possibilities are catered for. For example the
conjunction connective is written ‘ϙ’. If one
connective H is contained geometrically in another G, the implication
G(pq) → H(pq) holds, and merging two connectives
gives a connective equivalent to their conjunction. While this notation
is systematic and elegant, it has never caught on. Like
Łukasiewicz, Leśniewski always puts the connective before its
arguments, but he encloses the arguments in parentheses, thus the
conjunction of p and q is written
‘ϙ(pq)’. The reason for parentheses, which
Łukasiewicz could dispense with, will become clear. In this
“official” notation, the only quantifier is the universal,
written by placing the variables bound between lower corners, as for
example ‘ ˻pqf˼ ’, and placing the
quantifier scope or matrix within upper corners, so for example
‘˻pqf˼
⌈ϙ(f(pq)f(qp))⌉’
is Leśniewski's way of writing what we write as
‘∀pqf⌈f(pq)
∧ f(qp)⌉’. In his everyday
logical work and derivations however, Leśniewski used a slightly
modified version of Whitehead and Russell's Principia
Mathematica notation.

Among the variant versions of Protothetic Leśniewski mentions in
his 1929 article, one is a single-axiom version containing 82 signs.
(In 1945 Sobociski produced a single axiom version with 54 signs
only.) The other and more interesting idea is an algorithmic or
“computative” system of Protothetic, mentioned above in
Section 8. This is in effect a way to formalize the idea of
truth-tables, but applied beyond truth-functions of sentences to more
complex functors with arguments of first and higher order
truth-functions. Leśniewski does not take the idea very far, but
it was subsequently developed further by Owen Le Blanc (1991).

To modern students of logic accustomed to streamlined methods of
working with propositional logic, Leśniewski's Protothetic,
especially in its “official” versions, must appear very
cumbersome and difficult to understand and work with. In part this is a
question of the age of the systems and Leśniewski's aversion
to semantics (see above). However, Leśniewski did not
always present his work so fiercely. For purposes of everyday
derivation he employed what can only be described as a system of
natural deduction, making assumptions, following their consequences,
collecting them and inferring conditionals and biconditionals, in the
manner to which all students of logic have since become accustomed.
Astonishingly, neither he nor any of his students thought fit to codify
these practices into a system of rules. This was done instead at
Łukasiewicz's suggestion by Stanisław Jaśkowski.
The discovery of natural deduction is generally attributed to others,
but there is a chance that Leśniewski used it, in a recognizably
modern form, earlier than others. That he did not codify it probably
resulted from his viewing it as a pedagogical device and a way of
sketching out how a “proper” (i.e. axiomatic) proof was to
go. Another way to take some of the forbidding appearance out of
Protothetic is to look for more readily comprehensible axioms. The
following two-axiom set based on implication as the primitive (an
alternative Leśniewski also looked at) is rather straightforward;
the result was again obtained by Tarski:

P3

∀pq⌈p → (q →
p)⌉

P4

∀pqrf⌈f (rp) →
(f (r (p →
∀s⌈s⌉) ) → f (rq) )
⌉

The first is the universal closure of a standard axiom of
propositional calculus going back to Frege. Recalling that F(alse) can
be defined as ∀s⌈s⌉ and negation as
p → F, the second is the universal closure of

f (rp) → (f (r ~p)
→ f (rq))

which simply says that if f (r any truth-value) then
f(rq) for arbitrary q, and that it
obviously correct. It can be (tediously) checked that the result is
valid for all sixteen extensional binary truth-functions substitutible
for ‘f ’. Dragging the whole of Protothetic out
of these simple beginnings is obviously much harder, and depends on
coming up with suitable definitions for the connectives.

In Ontology, Leśniewski and his students, particularly
Sobociński, worked on replacing the 1920 “long”
single axiom by a shorter one, eventually arriving at the
unshortenable

OS

∀Aa⌈A ε a ↔
∀B⌈A ε B ∧
B ε a⌉⌉

This axiom gives a much less obvious insight into the intended
meaning of ‘ε’ than the original 1920 axiom. For a
delicately balanced combination of brevity with clarity the following
equivalent two-axiom set is perspicuous:

OS1

∀Ab⌈A ε b →
A ε A⌉

OS2

∀ABc⌈(A ε B ∧
B ε c) → A ε
c⌉

where notably the first axiom is precisely the one that
Leśniewski mentioned to Twardowski in 1919.

Although Ontology is perhaps the most generally interesting of
Leśniewski's systems, it was known during his lifetime less
through his own published work, which was confined to a short,
technical and inaccessible memoir, but through a gentle and
sympathetic exposition in Kotarbiński's 1929 widely read and
influential Warsaw texbook known simply as the
Elementy. Kotarbiński explains how he had no need to
come up with a logical system of names and predicates, since he could
get one ready-made from a firm with an excellent
reputation. Leśniewski was duly grateful for the plug.

The fact that the basic sentence-module in Ontology is singular
inclusion of the form ‘A ε b’
has misled some commentators into thinking that Leśniewski turned
his back on the Fregean notion of predication as functional
application and instead reverted to a medieval “two-name”
account of predication. Indeed, except for the matter of tense,
Leśniewski's account of the truth-conditions of such singular
sentences—that they are true if and only if the subject term
denotes a single object and the predicate term denotes one or more
objects of which this is one—is almost exactly the same as that
given by the medieval nominalist William of Ockham. However, whether
or not Ockham was a two-name theorist, Leśniewski definitely was
not. The general form of the singular sentence is the same as that of
any binary predication, f(ab), or in
Leśniewski's notation, f{ab}. Singular
inclusion is no syncategorematic copula: it is a special binary
predicate. That it is chosen as the primitive is understandable but
not compulsory. Leśniewski knew that other predicates than
‘ε’ could be taken as primitive, a fact emphasized
later by Lejewski.

It was in Mereology, Leśniewski's oldest system, that there were
the most varied developments. The 1927–30 article series on
Mereology as part of the foundations of mathematics rang the changes
in possible primitives. After scathing attacks on Whitehead's and
Russell's use/mention confusions in Principia, and on
standard set theories, he retraces the formal development of his 1916
paper, noting in a long footnote the similarities with Whitehead's
theory of events, whose formal development he also criticises. He
then summarizes the development up to 1920, continuing with additional
results unpublished in 1916, taking the number of theorems up to
198. Further chapters tidy up the axiomatization in terms of
‘part’, and show that ‘ingredient’ can be
taken as primitive. The theorems are taken up to 264, then
‘exterior’ is shown to be a possible primitive. There the
development is arrested, and a final section discusses singular
propositions of the form ‘A ε
b’, with a note on how to understand propositions about
a thing which changes. Using the example ‘Warsaw of 1830 is
smaller bigger than Warsaw of 1930’, Leśniewski proposes to
consider ‘Warsaw in 1830’ and ‘Warsaw in 1930’
as denoting time-slices of the temporally much longer object he calls
‘Warsaw from the beginning to the end of its
existence’. In this way he claims to bring many uses of
‘is’ in ordinary language within the scope of an analysis
using his Ontology. This four-dimensional understanding of ordinary
objects is now commonplace, but at that time it was something of a
rarity. The discussion is one of the few places in Leśniewski's
mature work where he indulges in anything similar to the philosophical
logic of his early years. Otherwise, when not discussing formal
systems and proving theorems, his prose discussions tend to be
intemperate though frequently justified criticisms of the statements
of others, most especially the proponents of standard set theory.

Normally the character of an academic is of marginal relevance to his
or her work. In the case of Leśniewski there is reason to think
otherwise. The extreme rigor that he applied in logic, the
unflichingly high standards he set himself and others, his blank
incomprehension of intellectual, formal and linguistic inexactness,
and his willingness to let academic disagreement sour his
relationships with colleagues, all speak of an unusual rigidity. This
seems to have been deeply anchored: of his school days we know little
other than that he was intolerant of exceptions to any rule, whether
or not the rule was sensible. After one of his early Polish essays was
typeset with a spelling mistake in the title
(‘środku’ instead of ‘środka’), he
always quoted the incorrect title, because to correct it would have
been to break the rule that quotation must be literal and exact. In
his early years he nursed the project of translating Anton Marty's
rambling and polemical 1908 treatise Untersuchungen zur
Grundlegung der allgemeinen Grammatik und Sprachphilosophie
[Investigations on the Founding of General Grammar and Philosophy
of Language]. He never got beyond the second word of the title,
‘zur’, which is admittedly not easily captured in all its
nuances—it can mean ‘toward’ as well as
‘on’ or ‘about’. Having carried a copy of the
book around for some time and asked all his friends and colleagues how
they would translate ‘zur’, he gave up. No doubt his
interests shifted, but the incident demonstrates both his
punctiliousness and his inflexibility.

Biographical material about Leśniewski is fairly sparse, and it
is even harder to gain a clear idea of what he was like as a person.
A photograph of him in Twardowski's seminar in 1913 shows a short,
dapper man with a goatee and a flamboyant neckscarf, as described by
Łukasiewicz. A later photograph from the same period lacks the
goatee but retains a moustache. The two known later photos show a
stiffly-posed, stocky, clean-shaven man in a business suit with the
slicked-back hair of the period, looking more like a bank manager than
a professor of logic, except for an intense stare. Leśniewski was
known to be a fierce critic of what he considered unclear, which was
pretty well everything. His stock complaint was that he could not
understand what speakers were saying, or writers were writing. In the
light of his pathological inability to see beyond the literal meaning
on the page to any intended but inaccurately expressed meaning this is
unsurprising, but also unendearing. In the early 1920s Marjan
Borowski, the editor of
Przegląd Filozoficzny, complained to Twardowski that
people were afraid to submit papers or give talks in Warsaw because
they were fearful of being criticised by Leśniewski, though he
added with some glee that the scourge of God had risen up against him
in the shape of one Tajtelbaum – the young Tarski. Even the
phlegmatic Twardowski found his former student irritating: in a diary
entry for 12 August 1930 he complains, “In general those who
behave according to Lesniewski's model ask very arbitrarily for
analysis where it is convenient to them—if, however, one of them
asks for analysis where it is not convenient to him, they turn to
intuition. And if the opposer in the discussion sometimes tries to
turn to intuition, they reply ‘We don’t understand what
you consider to be intuitively given’.” Twardowski's 1921
paper ‘Symbolomania and Pragmatophobia’ is a plea for
philosophers not to put symbols above things, and is clearly directed
against Łukasiewicz and Leśniewski and their students.

Leśniewski was however not lacking in a certain heavy humor.
Lejewski reported that he once mocked a Warsaw Professor of Classics
for wearing sunglasses (then rather rare): “Is the world too dazzling for
him?” He was equably resigned to his lectures being sparsely
attended because of their extreme technicality. One semester,
unexpectedly many students turned up at the first lecture. He looked
around the room in surprise and asked “What are you all doing
here? I am not Bergson.” For those who were just there for the
sake of doing a course and clocking up their attendance he quietly
signed off their lecture books straight away, and told them not to
worry about coming again. The few die-hards came for the sake of the
logic. Leśniewski would enter the lecture room with a briefcase
stuffed with papers, root around, find where he had got to and
carry on, writing formulas and explaining how they were derived. When
Quine visited some of these lectures, he was able to follow them
despite knowing no Polish.

Leśniewski married in 1913: his wife Zofia Prewysz-Kwinto came
from a landed family in Lithuania. They had no children. Before World
War I Leśniewski seems to have had the means to travel around to
different German cities to study, and to spend time after his doctorate
in Paris, San Remo and St. Petersburg.

Leśniewski inspired devotion in a very small coterie of students,
a few of whom stayed fiercely loyal, but sooner or later he ended up
alienating nearly everyone, either through his robust professional
opinions, or his manner, or his political views. He started out as a
radical socialist—his decision to spend the war in Russia was
partly personal and partly political—but after the excesses of
the October Revolution and its aftermath he rejected socialism. From
the 1920s he supported the authoritarianism of Józef
Piłsudski, but from about 1930 his views took on a darker
anti-Semitic tinge. An unsavory letter written to Twardowski in 1935
complains of “filthy tricks” being played on him “by
certain Jew-boys or their foreign friends”, declares a personal
antipathy to Tarski, whose career he would not obstruct, but admits he
“would be extraordinarily pleased if some day I were to read in
the newspapers that he was being offered a full professorship, for
example in Jerusalem, from where he could send us offprints of his
valuable works to our great profit.” Tarski had been passed over
for the chair in Lwów, which went to Chwistek on the strength of
praise from Russell, even though the Varsovians, including
Leśniewski, had supported Tarski. Tarski certainly felt aggrieved
and no doubt suspected anti-Semitic motives, but like Leśniewski
he was prickly and sensitive about matters of priority. It is in
retrospect mildly grotesque to see how carefully Leśniewski and
Tarski dance around each other in their prefaces and acknowledgements,
neither wishing to give public offence to the other. Despite their
disagreements and suspicions however, they long continued to meet once a week,
with no one else present, to discuss logic.

Leśniewski's antipathy towards set theory was so vehement and
his criticisms so intemperate that it led to a break in relations with
his set-theoretically minded mathematical colleagues Sierpiński
and Kuratowski; he resigned from the editorial board of Fundamenta
Mathematicae, with the result that he was no longer able to
publish his own work there. By the end of his life, Leśniewski's
only surviving close friend was the saintly patient and faithful
Kotarbiński, who alone of his colleagues visited him in hospital
in his last illness. They had been born just one day apart in
1886. The cancer that killed Leśniewski was no doubt exacerbated
by smoking: cigars and a large pipe. During the operation he was
conscious and without anaesthetic because of the dangers and he was
allowed to smoke even then to distract himself from the pain. But he
did not recover, and died sitting in his favourite armchair, brought
specially into the hospital.

Of the students that Leśniewski had taught at Warsaw, some went
on to logical and philosophical careers of their own, most notably
Tarski, whom Leśniewski correctly recognized as a genius, and who
went on to outshine his teacher. Among those who stayed fairly close
to Leśniewski's own views were Jerzy Słupecki, Bolesław
Sobociński, Czesław Lejewski, and Henry Hiż. The first
three in particular contributed after World War II to the
reconstruction of many of the logical results that had been lost in
1944. Serendipitously recovered students’ notes from
Leśniewski's lectures, translated and published in 1988, give
some idea of the detail of Leśniewski's teaching, but his
lectures ranged more widely than the extant works show. At all times
however, Leśniewski's logical position was a minority one, which
was respected while being rejected, and it won few converts after his
death. The reasons for his sidelining have been analysed by
Grzegorczyk (1955). Leśniewski's work was developed in the 1920s
when an axiomatic approach was standard, as in Hilbert's school, but
his negative attitude to semantics as a separable part of logic meant
he was unsympathetic to the shift towards a semantic approach
initiated by Tarski, and he would have remained opposed even had he
lived longer. His vituperative rejection of set theory made him no
friends and many enemies among mathematicians, while his unwillingness
and inability to find anything worthwhile in the works of philosophers
lost him their sympathy. His obsessive concern for the minute detail
of axiomatizations was unattractive to many when more streamlined
methods were available, and his radical nominalism made the
presentation of logic according to his principles a matter of extreme
inconvenience. Even Tarski, who was initially very much a follower,
had to admit that Leśniewski's conception of logical systems as
concrete collections of inscriptions growing in time through the
addition of new theses made them “thoroughly unrewarding objects
for methodological and semantic research”.

In retrospect we can see Leśniewski's obsession with the
fine detail of axiomatics and his rejection of semantics as conditioned
by his own idiosyncratic development and the predominant research
interests of the 1910s and 1920s. It is in fact possible to apply more
standard metalogical considerations to his systems, such as
investigating them for consistency and completeness. However, the
difficulties and complications of working within the confines of a
fully nominalistic attitude—sans sets, sans
abstract expression types—have put off all but a very few, and
the relative ease with which results can be obtained when fewer
ontological scruples are in play makes Leśniewski's systems
and others like them interesting mainly to certain philosophers, while
mathematicians and mathematical logicians have by-passed them.

On the other hand, Leśniewski's concerns for the proprieties
of quotation and use/mention, the object-language/metalanguage
distinction, canons of correct definition, and his development of
Mereology as the dominant formal theory of part and whole have all
passed into the mainstream, and the logical expertise he helped to
instil into a generation of Poles contributed in large part to making
Warsaw the premier interbellum location for mathematical logic. In the
entrance to the 1999 University of Warsaw Library building stand four
concrete pillars and sculptures by Adam Myjak, celebrating the
philosophical achievements of Poland. The figures represented are
Leśniewski's teacher Twardowski, his colleague
Łukasiewicz, his student Tarski, and Leśniewski himself.